1. Statistical Analysis of Stochastic Multi-Robot Boundary Coverage GANESH P KUMAR & SPRING M BERMAN FULTON SCHOOLS OF ENGINEERING ARIZONA STATE UNIVERSITY 1

2. Why Stochastic Boundary Coverage? 2 Distributed sensing J. Brandon, Digital Trends, 7/13/11 http://goo.gl/utjpbc Targeted drug delivery Sinha et al., Mol. Cancer. Ther. 06 http://goo.gl/zXVHGw Collective transportImaging cancer cells Qin et al., Adv. Funct. Mater. 12 http://goo.gl/ZhRYSb Nanoscale Applications Macroscale Applications 3. What is Stochastic Boundary Coverage? Robots: Occupy random positions along boundaries Sense or communicate within a local range Dont have: Global position Map of environment Multi-Robot Boundary Coverage [T. Pavlic, S. Wilson, G. Kumar, and S. Berman, ISRR 13] 3 Example Goal: Achieve target allocation on boundary 4. Related Work 4 Design of collective behaviors in robotic swarms Active Self-Assembly [Napp, Burden, & Klavins, RSS06, RSS09] Product Assembly [Matthey, Berman, & Kumar, ICRA09] Chain Formation [Evans, Mermoud, & Martinoli, ICRA10] Distributed Manipulation [Martinoli, Easton, & Agassounon, IJRR 04] Turbine Inspection [Correll & Martinoli, DARS 07] Macroscopically model collective behavior Require prior knowledge of encounter rates 5. Related Work 5 Models of adsorption processes Langmuir adsorption (reversible) [Langmuir, J Amer Chem Soc, 1918] Random sequential adsorption (irreversible) [Talbot et al., Colloid Surface A 00] Robot attachments may be modelled as adsorption Attachment rates controlled to get desired coverage 6. Saturation: Densest Possible Stochastic Coverage CLOSED BOUNDARY Distance OPEN BOUNDARY 6 7. Problem Statement Let robots of diameter 2 attach at uniformly random positions to a unit- length closed boundary. Given saturating distance , find: Probability of Saturation Distribution of robot positions Distribution of inter-robot distances 7 Not saturated Saturated n = 14 8. Agenda Solve problem for point robots ( = 0) Explain extension for finite-sized robots ( > 0) Validate results with Monte Carlo simulations 8 Note: Well consider only closed unit-length boundaries from now on. 9. Point Robots: Saturation CLOSED BOUNDARY 1 2 1 2 1 = 0 +1 = 1 Open boundary: = [, ] Robot attaches at with 1 = 0 and +1 = 1 Saturation implies +1 , = 1,2, , 9 10. Point Robots: Saturation Here is called the -th order statistic of a uniform parent (2, 3, , ) is a point in the event space Since +1 , valid configurations lie in the event simplex n = 2, 3, , : 0 +1 1 1 1 n = 1 1 ! 3 (0,0) (0,1) (1,0) 3 2 10 11. Concept of Slack Define Total slack = length of curve = 1 Individual slack = +1 Individual slacks (1, 2, , ) determine slack space 1 2 3 = 1 1 + 2 + 3 = 11 12. Geometric Interpretation of Valid slacks lie on the regular slack simplex = { 1, 2, , : 0 , = } 1 Saturated configurations lie in the saturated hypercube: = { 1, 2, , : 0 } We have = ( ) 3 (0,0,0) (, , ) 3 (0, , 0) (, 0,0) (0,0, ) 2 12 13. Computing Simplex Volume The volume of dimensional regular simplex of side is () = 2 . +1 ! Since is regular with side 2, 1 = 1. 1 ! E.g. 2 = 2 3 2 = 1 ( ) 13 14. Computing Intersection Volume Not straightforward at all, requires indirect attack! Let be a -element subset of slack axes {1, 2, , }. Consider a sub-region of on which slacks in are unsaturated. ( ) = 1, 2, , : Then ( ) is a regular simplex ! 1 ( ) = 1 1 ! = 1 1( ) 14 15. Computing Intersection Volume Note that unsaturated region = ( ) for all nonempty {1, 2, , } Its volume can be computed by the Inclusion-Exclusion Principle 1 2 = + + 1 (1 2 ) Writing in this form gives our result! = ( ) where = . 15 = 1 1 1( ) 16. Point Robots: PDFs of Robot Positions Joint pdf of robot positions is the uniform pdf 2, 3, , = 1 ! To get (), marginalize over remaining statistics by repeated integration = 1 1 , = 2,3, , This is a Beta density, and simplifies to: = (| 1, + 1 ) 16 , = + 1 ! 1 ! 1 ! 1 1 1 17. Point Robots: PDF of Slacks Define the domain comprised by all but the last slack: D { 1, 2, , 1 : 0 1} The joint pdf of slacks is found to be uniform over D: 1, 2, , 1 = 1 ! D Marginalize over remaining slacks to get (): = (|1, 1) 17 18. Extension to Finite-Sized Robots Robots attaching sequentially at uniformly random positions result in Renyis Parking Problem scenario [Dvoretzky & Robbins 64] Resembles adsorption of particles on a surface We solve a different problem which extends point-robot analysis Robots coordinate to avoid overlaps using Slack Attach protocol 18 19. Finite-Sized Robots: Slack Definitions Total Slack = length of boundary unoccupied after robots attach: 1 2 Individual slack = closest distance between adjacent robots: = +1 2 1 2 2 = 1 2 +1 19 20. Slack Attach Protocol Boundary 1 Step 1 All robots attach adjacent to one another. The most clockwise one is called 1 20 21. Slack Attach Protocol Boundary Step 2 First robot detaches and travels around boundary to measure , then reattaches to original position 1 21 22. Slack Attach Protocol Step 3 Robots collectively choose 1 1 uniformly randomly from [0, ] and sort it in increasing order. They compute individual slacks = + 1 [] [1]0 [2] [3] 1 2 3 4 22 23. Slack Attach Protocol Boundary Step 4 Robots choose positions = 1 + 1 + 2 with 1 0 1 2 3 4 23 24. Finite-Sized Robots: Saturation Saturation = no robot can enter between two adjacent ones 4 2 Simplex-Hypercube intersection gives us = 1 =1 1 1( ) 1 2 1 where = 12 2 24 25. Finite-Sized Robots: Robot Positions & Slacks Slacks have the scaled Beta pdf = . (|1, 1) Order statistics non-trivial! 1 = 0 2 = . 1, 1 + 2 3 = 2 + 2 + 2 We dont know correlations between 2 and 2 nor the joint pdf of order statistics! 25 26. Results: Comparison with Monte Carlo Simulations Analytical solutions for compared with averages of 20,000 Monte Carlo Trials Analytical solutions for order statistics and slack densities compared with averages of 5,000 Monte Carlo Trials 26 27. Validating for point robots 27 Analytical Monte Carlo as , # of robots Sat. distance 28. Validating for finite-sized robots 28 Analytical Monte Carlo # of robots Robot radius as , ; high (, ) unphysical 29. Validating Robot Position PDF for Point Robots Frequency plot of 5000 samples of 2 fit to a (|1,4) density for = 5 point robots 29 531 = 0 6 = 14 30. Validating Slack PDF for Point Robots Frequency plot of 5000 samples of 2 = 3 2 fit to a (|1,4) density for = 5 point robots 30 1 = 0 2 53 6 = 14 31. Conclusion Posed the problem of computing saturation probability Analytically determined and order statistics for point robots Extended analysis to finite sized robots, explaining limitations Validated results with Monte Carlo simulations 31 32. Future Work Extend finite-sized robots case to asynchronous attachment Attachment over time, without coordination Enable analysis of our prior simulations [ISRR 13] Determine order statistics and slack pdfs given saturation sat , (|sat) Develop distributed controllers for multi-robot transport Goal : achieve robustness of ant food retrieval 32 [G. Kumar, A. Buffin, T. Pavlic, S.Pratt, and S. Berman, HSCC13] 33. Acknowledgements Sean Wilson, Theodore Pavlic, Ruben Gameros : for valuable discussions on paper and presentation 33 The Autonomous Collective Systems Lab Front: Prof. Spring Berman From left to right: Dr. Theodore Pavlic (Postdoc collaborator) Ruben Gameros (Masters student) Karthik Elamvazhuthi (Masters student) Ganesh Kumar (PhD student) Sean Wilson (PhD student)